Теория возможностей
Теория возможностей — математическая теория, имеющая дело с особым типом неопределенности, альтернативна теории вероятностей. Профессор Лотфи Заде (Lotfi Zadeh) впервые ввел теорию возможностей в 1978 в качестве расширения его теорий нечетких множеств и нечеткой логики. Д.Дюбуа (D. Dubois) и Г.Праде (H. Prade) позже внесли свой вклад в ее развитие. Раньше, в 50-х экономист Дж. Л. С. Шакле (G.L.S. Shackle) предложил min/max алгебру для описания степени потенциальных неожиданностей. В конце 90-х профессор МГУ Пытьев Ю. П. предложил вариант теории возможностей, в котором возможность и необходимость определяются значениями линейного счётно-аддитивного функционала (интеграла). Содержательное толкование теоретико-возможностных методов существенно отличается от теоретико-вероятностных. Здесь возможность события, в отличие от вероятности, которая оценивает частоту его появления в регулярном стохастическом эксперименте, ориентирована на относительную оценку истинности данного события, его предпочтительности в сравнении с любым другим. То есть содержательно могут быть истолкованы лишь отношения «больше», «меньше» или «равно». Вместе с тем возможность не имеет событийно-частотной интерпретации (в отличие от вероятности), которая связывает её с экспериментом. Тем не менее теория возможностей позволяет математически моделировать реальность на основе опытных фактов, знаний, гипотез, суждений исследователей. Формализация возможности For simplicity, assume that the universe of discourse Ω is a finite set, and assume that all subsets are measurable. A distribution of possibility is a function \operatorname{pos} from Ω to 1 such that: : Axiom 1: \operatorname{pos}(\varnothing) = 0 : Axiom 2: \operatorname{pos}(\Omega) = 1 : Axiom 3: \operatorname{pos}(U \cup V) = \max \left( \operatorname{pos}(U), \operatorname{pos}(V) \right) for any disjoint subsets U and V . It follows that, like probability, the possibility measure is determined by its behavior on singletons: : \operatorname{pos}(U) = \max_{\omega \in U} \operatorname{pos}(\{\omega\}) Axiom 1 can be interpreted as the assumption that Ω is an exhaustive description of future states of the world, because it means that no belief weight is given to elements outside Ω. Axiom 2 could be interpreted as the assumption that the evidence from which \operatorname{pos} was constructed is free of any contradiction. Technically, it implies that there is at least one element in Ω with possibility 1. Axiom 3 corresponds to the additivity axiom in probabilities. However there is an important practical difference. Possibility theory is computationally more convenient because Axioms 1-3 imply that: : \operatorname{pos}(U \cup V) = \max \left( \operatorname{pos}(U), \operatorname{pos}(V) \right) for any subsets U and V . Because one can know the possibility of the union from the possibility of each component, it can be said that possibility is compositional with respect to the union operator. Note however that it is not compositional with respect to the intersection operator. Generally: : \operatorname{pos}(U \cap V) \leq \min \left( \operatorname{pos}(U), \operatorname{pos}(V) \right) Remark for the mathematicians: When Ω is not finite Axiom 3 can be replaced by: : For all index sets I , if the subsets U_{i,\, i \in I} are pairwise disjoint, \operatorname{pos}\left(\cup_{i \in I} U_i\right) = \sup_{i \in I}\operatorname{pos}(U_i) Необходимость Whereas probability theory uses a single number, the probability, to describe how likely an event is to occur, possibility theory uses two concepts, the possibility and the necessity ''of the event. For any set U , the necessity measure is defined by : \operatorname{nec}(U) = 1 - \operatorname{pos}(\overline U) In the above formula, \overline U denotes the complement of U , that is the elements of \Omega that do not belong to U . It is straightforward to show that: : \operatorname{nec}(U) \leq \operatorname{pos}(U) for any U and that: : \operatorname{nec}(U \cap V) = \min ( \operatorname{nec}(U), \operatorname{nec}(V)) Note that contrary to probability theory, possibility is not self-dual. That is, for any event U , we only have the inequality: : \operatorname{pos}(U) + \operatorname{pos}(\overline U) \geq 1 However, the following duality rule holds: : For any event U , either \operatorname{pos}(U) = 1 , or \operatorname{nec}(U) = 0 Accordingly, beliefs about an event can be represented by a number and a bit. Интерпретация There are four cases that can be interpreted as follows: \operatorname{nec}(U) = 1 means that U is certainly true. It implies that \operatorname{pos}(U) = 1 . \operatorname{pos}(U) = 0 means that U is certainly false. It implies that \operatorname{nec}(U) = 0 . \operatorname{pos}(U) = 1 means that I would not be surprised at all if U occurs. It leaves \operatorname{nec}(U) unconstrained. \operatorname{nec}(U) = 0 means that I would not be surprised at all if U does not occur. It leaves \operatorname{pos}(U) unconstrained. The intersection of the last two cases is \operatorname{nec}(U) = 0 and \operatorname{pos}(U) = 1 meaning that I believe nothing at all about U . Because it allows for indeterminacy like this, possibility theory relates to the graduation of a three-valued logic, such as intuitionistic logic, rather than the classical two-valued logic. Note that unlike possibility, fuzzy logic is compositional with respect to both the union and the intersection operator. The relationship with fuzzy theory can be explained with the following classical example. * Fuzzy logic: When a bottle is half full, it can be said that the level of truth of the proposition «The bottle is full» is 0.5. The word «full» is seen as a fuzzy predicate describing the amount of liquid in the bottle. * Possibility theory: There is one bottle, either completely full or totally empty. The proposition «the possibility level that the bottle is full is 0.5» describes a degree of belief. One way to interpret 0.5 in that proposition is to define its meaning as: I am ready to bet that it’s empty as long as the odds are even (1:1) or better, and I would not bet at any rate that it’s full. Теория возможностей как уточнение теории вероятностей * There is an extensive formal correspondence between probability and possibility theories, where the addition operator corresponds to the maximum operator. * A possibility measure can be seen as a consonant plausibility measure in Dempster-Shafer theory (Dempster-Shafer theory) of evidence. The operators of possibility theory can be seen as a hyper-cautious version of the operators of the transferable belief model, a modern development of the theory of evidence. * Possibility can be seen as an upper probability: any possibility distribution defines a unique set of admissible probability distributions by :: \left\{\, p: \forall S, p(S)\leq \operatorname{pos}(S)\,\right\}. This allows one to study possibility theory using the tools of imprecise probabilities. Ссылки * Dubois, Didier and Prade, Henri, «Possibility Theory, Probability Theory and Multiple-valued Logics: A Clarification», ''Annals of Mathematics and Artificial Intelligence 32:35-66, 2001. * Zadeh, Lotfi, «Fuzzy Sets as the Basis for a Theory of Possibility», Fuzzy Sets and Systems 1:3-28, 1978. (Reprinted in Fuzzy Sets and Systems 100 (Supplement): 9-34, 1999.) * Пытьев Ю.П., «Возможность. Элементы теории и применения», Эдиториал УРСС, 2000. См. также * Логическая вероятность * Теория вероятностей * Теория нечеткой меры * Верхние и нижние вероятности * Эвентология * Теория очевидностей Демпстера-Шафера * Нечёткая логика * Копула Категория:Нечёткая логика